The spectrum of the preconditioned matrix \({{\bf P}}^{-1}{\bf A}\)  is uniformly bounded below by \(1/2\)  and bounded above by \(3/2\) .


Let \({{{\bf z}}}\in {{\mathbb {R}}}^{N}\)  be any nonzero vector. By the Rayleigh quotients theorem (see Theorem 4.2.2 in [1]}) and lmbda-PA, for each \(i\) , it holds
\(\frac{1}{2}<{\lambda }_{\min }({\tau (A_{i})^{-1}A_{i}})\le \frac{{{{\bf z}}}^{\intercal }A_{i}{{{\bf z}}}}{{{{\bf z}}}^{\intercal }\tau (A_{i}){{{\bf z}}}}\le {\lambda }_{\max }({\tau (A_{i})^{-1}A_{i}})<\frac{3}{2},\) 
